Module d'élasticité isostatique

Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir plus.

Cet article concernant la physique doit être recyclé (mai 2015).

Une réorganisation et une clarification du contenu paraissent nécessaires. Améliorez-le, discutez des points à améliorer ou précisez les sections à recycler en utilisant {{section à recycler}}.

Le module d'élasticité isostatique^[1] (en anglais : bulk modulus) est la constante qui relie la contrainte au taux de déformation d'un matériau isotrope soumis à une compression isostatique.

Expression

Généralement noté $K$ ( $B$ en anglais), le module d'élasticité isostatique permet d'exprimer la relation de proportionnalité entre le premier invariant du tenseur des contraintes et le premier invariant du tenseur des déformations :

Module d'élasticité isostatique de quelques matériaux
Air	101 kPa (isotherme) (142 kPa en adiabatique)
Eau	2,2 GPa (augmente avec la pression)
Verre	35 à 55 GPa
Acier	160 GPa
Diamant	442 GPa

s=K\,e

où :

$s=\sum _{i}{\frac {1}{3}}\sigma _{ii}$ est la contrainte isostatique (en unité de pression) ;
$K$ est le module d'élasticité isostatique (en unité de pression) ;
$e=\sum _{i}\varepsilon _{ii}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}$ est le taux de déformation isostatique^[2] (sans dimension).

Il s'exprime, respectivement vis-à-vis des coefficients de Lamé ou du module de Young et du coefficient de Poisson, par :

K=\lambda +{\frac {2}{3}}\,\mu ={\frac {1}{3}}\,{\frac {E}{(1-2\nu )}}

Notes :

pour $\nu =1/3$ , $K=E$ ;
pour $\nu \to 1/2$ , $K\to +\infty$ (incompressibilité).

Les matériaux métalliques sont proches du premier cas ( $K\approx E$ dans leur domaine élastique) alors que les élastomères s'approchent d'un comportement incompressible ( $K\gg E$ ).

On peut aussi exprimer $K$ en fonction des modules d'élasticité en traction $E$ et en cisaillement $G$ :

{\frac {1}{K}}={\frac {9}{E}}-{\frac {3}{G}}

Le module d'élasticité isostatique représente la relation de proportionnalité entre la pression et le taux de variation du volume :

\Delta P=-K\,{\frac {\Delta V}{V_{0}}}

C'est l'inverse de la compressibilité isotherme $\chi _{T}$ , définie en thermodynamique par :

{\frac {1}{K}}=\chi _{T}=-{\frac {1}{V}}\,\left({\frac {\partial V}{\partial P}}\right)_{T}

Notes et références

↑ Synonymes : module d'élasticité à la compression isostatique, module de rigidité à la compression, module d'élasticité cubique, module d'incompressibilité, module de compression hydrostatique, module de dilatation volumique, module d'élasticité volumique, etc.
↑ Synonyme : taux de dilatation cubique.

Voir aussi

Bibliographie

P. Germain, Mécanique des milieux continus, 1962, Masson et Cie.
G. Duvaut, Mécanique des milieux continus, 1990, Masson.

Articles connexes

v · m

Modules d'élasticité pour des matériaux homogènes et isotropes

Module d'Young (E)
Module de cisaillement (G)
Module d'élasticité isostatique (K)
Premier coefficient de Lamé (λ)
Coefficient de Poisson (ν)
Module d'onde de compression (M, P-wave modulus)

Formules de conversion

Les propriétés élastiques des matériaux homogènes, isotropes et linéaires sont déterminées de manière unique par deux modules quelconques parmi ceux-ci. Ainsi, on peut calculer chacun à partir de deux d'entre eux en utilisant ces formules.

formules en 3D	$(\lambda ,G)$	$(E,G)$	$(K,\lambda )$	$(K,G)$	$(\lambda ,\nu )$	$(G,\nu )$	$(E,\nu )$	$(K,\nu )$	$(K,E)$	$(M,G)$
$K\,[\mathrm {Pa} ]=$	$\lambda +{\tfrac {2G}{3}}$	${\tfrac {EG}{3(3G-E)}}$			${\tfrac {\lambda (1+\nu )}{3\nu }}$	${\tfrac {2G(1+\nu )}{3(1-2\nu )}}$	${\tfrac {E}{3(1-2\nu )}}$			$M-{\tfrac {4G}{3}}$
$E\,[\mathrm {Pa} ]=$	${\tfrac {G(3\lambda +2G)}{\lambda +G}}$		${\tfrac {9K(K-\lambda )}{3K-\lambda }}$	${\tfrac {9KG}{3K+G}}$	${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}$	$2G(1+\nu )\,$		$3K(1-2\nu )\,$		${\tfrac {G(3M-4G)}{M-G}}$
$\lambda \,[\mathrm {Pa} ]=$		${\tfrac {G(E-2G)}{3G-E}}$		$K-{\tfrac {2G}{3}}$		${\tfrac {2G\nu }{1-2\nu }}$	${\tfrac {E\nu }{(1+\nu )(1-2\nu )}}$	${\tfrac {3K\nu }{1+\nu }}$	${\tfrac {3K(3K-E)}{9K-E}}$	$M-2G$
$G\,[\mathrm {Pa} ]=$			${\tfrac {3(K-\lambda )}{2}}$		${\tfrac {\lambda (1-2\nu )}{2\nu }}$		${\tfrac {E}{2(1+\nu )}}$	${\tfrac {3K(1-2\nu )}{2(1+\nu )}}$	${\tfrac {3KE}{9K-E}}$
$\nu \,[1]=$	${\tfrac {\lambda }{2(\lambda +G)}}$	${\tfrac {E}{2G}}-1$	${\tfrac {\lambda }{3K-\lambda }}$	${\tfrac {3K-2G}{2(3K+G)}}$					${\tfrac {3K-E}{6K}}$	${\tfrac {M-2G}{2M-2G}}$
$M\,[\mathrm {Pa} ]=$	$\lambda +2G$	${\tfrac {G(4G-E)}{3G-E}}$	$3K-2\lambda \,$	$K+{\tfrac {4G}{3}}$	${\tfrac {\lambda (1-\nu )}{\nu }}$	${\tfrac {2G(1-\nu )}{1-2\nu }}$	${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}$	${\tfrac {3K(1-\nu )}{1+\nu }}$	${\tfrac {3K(3K+E)}{9K-E}}$
formules en 2D	$(\lambda _{\mathrm {2D} },G_{\mathrm {2D} })$	$(E_{\mathrm {2D} },G_{\mathrm {2D} })$	$(K_{\mathrm {2D} },\lambda _{\mathrm {2D} })$	$(K_{\mathrm {2D} },G_{\mathrm {2D} })$	$(\lambda _{\mathrm {2D} },\nu _{\mathrm {2D} })$	$(G_{\mathrm {2D} },\nu _{\mathrm {2D} })$	$(E_{\mathrm {2D} },\nu _{\mathrm {2D} })$	$(K_{\mathrm {2D} },\nu _{\mathrm {2D} })$	$(K_{\mathrm {2D} },E_{\mathrm {2D} })$	$(M_{\mathrm {2D} },G_{\mathrm {2D} })$
$K_{\mathrm {2D} }\,[\mathrm {N/m} ]=$	$\lambda _{\mathrm {2D} }+G_{\mathrm {2D} }$	${\tfrac {G_{\mathrm {2D} }E_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}$			${\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}$	${\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}$	${\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}$			$M_{\mathrm {2D} }-G_{\mathrm {2D} }$
$E_{\mathrm {2D} }\,[\mathrm {N/m} ]=$	${\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}$		${\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}$	${\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}$	${\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}$	$2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,$		$2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })$		${\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}$
$\lambda _{\mathrm {2D} }\,[\mathrm {N/m} ]=$		${\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}$		$K_{\mathrm {2D} }-G_{\mathrm {2D} }$		${\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}$	${\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}$	${\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}$	${\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}$	$M_{\mathrm {2D} }-2G_{\mathrm {2D} }$
$G_{\mathrm {2D} }\,[\mathrm {N/m} ]=$			$K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }$		${\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}$		${\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}$	${\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}$	${\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}$
$\nu _{\mathrm {2D} }\,[1]=$	${\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}$	${\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1$	${\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}$	${\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}$					${\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}$	${\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}$
$M_{\mathrm {2D} }\,[\mathrm {N/m} ]=$	$\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }$	${\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}$	$2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }$	$K_{\mathrm {2D} }+G_{\mathrm {2D} }$	${\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}$	${\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}$	${\tfrac {E_{\mathrm {2D} }}{(1-\nu _{\mathrm {2D} })(1+\nu _{\mathrm {2D} })}}$	${\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}$	${\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}$

Read other articles:

Gedung Balai Budaya Pandeglang

Artikel ini perlu diwikifikasi agar memenuhi standar kualitas Wikipedia. Anda dapat memberikan bantuan berupa penambahan pranala dalam, atau dengan merapikan tata letak dari artikel ini. Untuk keterangan lebih lanjut, klik [tampil] di bagian kanan. Mengganti markah HTML dengan markah wiki bila dimungkinkan. Tambahkan pranala wiki. Bila dirasa perlu, buatlah pautan ke artikel wiki lainnya dengan cara menambahkan [[ dan ]] pada kata yang bersangkutan (lihat WP:LINK untuk keterangan lebih lanjut...

Pulau Keeling Utara

Peta Kepulauan Cocos (Keeling). Pada peta, Pulau Keeling Utara terletak paling atas dari kepulauan Pulau Keeling Utara Pulau Keeling Utara, dalam bahasa Inggris North Keeling Island, adalah sebuah pulau karang kecil tak berpenghuni. Luasnya sekitar 1,2 kilometer persegi. Berada pada jarak sekitar 25 kilometer atau 16 mil arah utara dari Pulau Horsburgh. Pulau ini adalah sebuah atol sekaligus pulau paling utara wilayah Australia, Kepulauan Cocos (Keeling). Bentuk Pulau Keeling Utara yang seper...

Asosiasi Negara-Negara Pesisir Samudra Hindia

Asosiasi Negara-Negara Pesisir Samudra Hindia (IORA)Peta negara-negara IORAMarkasEbene Cyber City, MauritiusBahasa kerjaInggrisTipeAntarpemerintahKeanggotaan 23 negara Afrika Selatan Australia Bangladesh India Indonesia Iran Kenya Komoro Madagaskar Maladewa Malaysia Mauritius Mozambik Oman Prancis Seychelles Singapura Somalia Sri Lanka Tanzania Thailand Uni Emirat Arab Yaman Pem...

Felice Piemontese

Felice Piemontese Felice Piemontese (Monte Sant'Angelo, 25 gennaio 1942) è uno scrittore, giornalista e poeta italiano. Indice 1 Biografia 2 Opere 2.1 Curatele 2.2 Narrativa 2.3 Poesia 2.4 Saggi 2.5 Traduzioni 3 Note 4 Bibliografia 5 Collegamenti esterni Biografia Nato nel 1942 a Monte Sant'Angelo, in provincia di Foggia, vive dal 1946 a Napoli. La famiglia vi si trasferì infatti alla fine della seconda guerra mondiale dal momento che l'attività sartoriale del padre era stata resa particol...

Orly Airport

Secondary airport serving Paris, France For the World War I and NATO military use of this facility, see Orly Air Base. Paris Orly AirportAéroport de Paris-OrlyIATA: ORYICAO: LFPOWMO: 07149SummaryAirport typePublicOwnerGroupe ADPOperatorParis AéroportServesParis metropolitan areaLocationEssonne and the Val-de-Marne, FranceOpened1932; 92 years ago (1932)Hub forAir FranceOperating base foreasyJetTransavia FranceVuelingBuilt1 January 1918; 106 years ago (1918...

Bibliography of William Howard Taft

This article is part of a series aboutWilliam Howard Taft Early life Family Bibliography Legacy 27th President of the United States Inauguration Presidency (timeline) Executive actions Foreign policy Taftian theory Domestic policy Cabinet Judiciary Dollar diplomacy Income Tax amendment Payne–Aldrich Tariff Act Wireless Ship Act of 1910 Mann–Elkins Act Defense Secrets Act Radio Act of 1912 Commission on Economy and Efficiency U.S. occupation of Nicaragua Presidential campaigns 1908 conven...

Propaganda komunis

Kamerad Lenin Membersihan Kotoran dari Dunia karya Viktor Deni. Jarum jam berlabel komunisme nyaris memotong kepala karikatur berhidup besar dan bertopi tinggi berlabel Kapital dengan keterangan jam akhir!. Propaganda komunis adalah promosi saintifik, artistik dan sosial terhadap ideologi komunisme, pandangan dunia komunis dan kepentingan gerakan komunis. Meskipun istilah tersebut memiliki konotasi negatif di dunia Barat, istilah propaganda banyak merujuk kepada publikasi atau kampanye apapun...

Andrew Moray

13th-century Scottish esquire For other people named Andrew Moray, see Andrew Moray (disambiguation). Andrew MorayPersonal detailsBornScotland, exact location of birth is not knownDied1297Cause of deathDue to wounds received at the Battle of Stirling BridgeChildrenSir Andrew MurrayParent(s)Sir Andrew Moray of Pettyan unnamed daughter of John Comyn I of BadenochRelativesDavid Moray (uncle)OccupationMilitary leaderMilitary serviceAllegianceKingdom of ScotlandYears of service1297R...

Galatasaray S.K. (women's football)

Football clubGalatasaray Spor KulübüFull nameGalatasaray Spor Kulübü (Galatasaray Sports Club)Nickname(s)Dişi Aslanlar (The Lionesses)Sarı Kırmızılılar (The Yellow Reds)Cimbom (The Cimbom)Short nameGSFounded2011GroundFlorya Metin Oktay FacilitiesPresidentDursun ÖzbekHead coach Metin ÜlgenLeagueTurkcell Women's Football LeagueWebsiteClub website Home colours Away colours Current season Active departments of Galatasaray S.K. Athletics Basketball(Men's) Basketball(Women's) Chess Bri...

Філософія

Частина серії проФілософіяLeft to right: Plato, Kant, Nietzsche, Buddha, Confucius, AverroesПлатонКантНіцшеБуддаКонфуційАверроес Філософи Епістемологи Естетики Етики Логіки Метафізики Соціально-політичні філософи Традиції Аналітична Арістотелівська Африканська Близькосхідна іранська Буддій�...

Nimitz High School (Irving, Texas)

Co-educational, public, secondary school in Irving, Dallas County, Texas, United StatesNimitz High SchoolAddress100 West Oakdale RoadIrving, Dallas County, Texas 75060United StatesInformationTypeCo-Educational, Public, SecondaryMottoExcellence and equity for all.Established1968School districtIrving Independent School DistrictPrincipalFrancisco MirandaStaff174.96 (FTE)[1]Grades9-12Enrollment2,523 (2019-20)[1]Student to teacher ratio14.42Color(s) Athletics con...

Edison Assembly

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: Edison Assembly – news · newspapers · books · scholar · JSTOR (September 2020) Edison AssemblyLocation in New JerseyOperated1948 (1948)–2004 (2004)LocationEdison, New JerseyCoordinates40°31′01″N 74°22′19″W / 4...

Pak Choong-hoon

Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Pak Choong-hoon – berita · surat kabar · buku · cendekiawan · JSTOR Artikel ini perlu dikembangkan agar dapat memenuhi kriteria sebagai entri Wikipedia.Bantulah untuk mengembangkan artikel ini. Jika tida...

Goda Butkutė

Lithuanian pair skater Goda ButkutėBorn (1999-07-10) 10 July 1999 (age 24)Kaunas, LithuaniaHeight1.53 m (5 ft 0 in)Figure skating careerCountryLithuaniaPartnerNikita ErmolaevCoachAlexander Smirnov, Lilija VanagienėSkating clubBaltu AiniaiBegan skating2003 Goda Butkutė (born 10 July 1999) is a Lithuanian pair skater. With her skating partner, Nikita Ermolaev, she won five senior international medals, including three on the ISU Challenger Series.[1] Personal life ...

Deryck Whibley

Canadian musician (born 1980) Deryck WhibleyWhibley performing in 2023BornDeryck Jason Whibley (1980-03-21) 21 March 1980 (age 44)Scarborough, Ontario, Canada[1]Other namesBizzy DGunnerEducationExeter High SchoolOccupations Musician singer songwriter record producer Spouses Avril Lavigne (m. 2006; div. 2010) Ariana Cooper (m. 2015)Children2Musical careerGenres Pop punk alternative rock...

Senate of Mauritania

Politics of Mauritania Member State of the Arab League Constitution Human rights Slavery Government President Mohamed Ould Ghazouani Prime Minister Mohamed Ould Bilal Parliament National Assembly President Administrative divisions Regions Departments Communes Elections Recent elections Presidential: 201420192024 Parliamentary: 201320182023 Political parties Electoral districts Foreign relations Ministry of Foreign Affairs Minister: Ismail Ould Cheikh Ahmed Diplomatic missions of / in Mauritan...

Shirley-Anne Somerville

Scottish Social Justice Secretary Shirley-Anne SomervilleMSPOfficial portrait, 2024Cabinet Secretary for Social JusticeIncumbentAssumed office 29 March 2023First MinisterHumza Yousaf John SwinneyPreceded byShona RobisonCabinet Secretary for Education and SkillsIn office20 May 2021 – 29 March 2023First MinisterNicola SturgeonPreceded byJohn SwinneySucceeded byJenny GilruthCabinet Secretary for Social Security and Older PeopleIn office26 June 2018 – 20 May 2021First Mi...

Lohuec

LohueccomuneLohuec – VedutaChiesa di Saint-Judoce a Lohuec LocalizzazioneStato Francia Regione Bretagna Dipartimento Côtes-d'Armor ArrondissementGuingamp CantoneCallac TerritorioCoordinate48°28′N 3°31′W48°28′N, 3°31′W (Lohuec) Altitudine210 m s.l.m. Superficie17,53 km² Abitanti276[1] (2009) Densità15,74 ab./km² Altre informazioniCod. postale22160 Fuso orarioUTC+1 Codice INSEE22132 CartografiaLohuec Modifica dati su Wikidata · Manua...

Grant County, Kentucky

County in Kentucky, United States County in KentuckyGrant CountyCountyThe Grant County historic courthouse in WilliamstownLocation within the U.S. state of KentuckyKentucky's location within the U.S.Coordinates: 38°38′N 84°37′W / 38.64°N 84.61°W / 38.64; -84.61Country United StatesState KentuckyFounded1820SeatWilliamstownLargest cityWilliamstownArea • Total261 sq mi (680 km2) • Land258 sq mi (670 k...

Special operations

Specially designated military operations that are considered unconventionalFor the Russian government euphemism for invading Ukraine, see Special military operation.Spec Ops redirects here. For the video game series, see Spec Ops (series). For the Indian web series, see Special OPS. For the forces that perform spec ops, see special forces. U.S. Air Force Special Tactics operators in MultiCam uniforms training in Jordan Special operations or special ops are military activities conducted, accor...

'Ndrangheta in provincia di Reggio Calabria

/profillengkap.com/index.php/article/Avraam Benaroya

Silvestrine

Kekaisaran Asyur Pertengahan

Kapuwila, Lalembuu, Konawe Selatan

Buranga, Ampibabo, Parigi Moutong

Eunidia duffyana

Chris McHugh

Waktu Timur

Wikipedia bahasa Belanda

1922 New England Textile Strike

Buduk Kubul, Krayan Barat, Nunukan